You have seven balls labelled 1,2,…,7 and seven unlabelled, indistinguishable boxes. How many different ways can the balls be placed into the boxes? Can anyone please work the complete solution. I got the answer but it was solved by my brother and he challenged me to get the complete solution ... Please ...

Answer 1

The boxes are indistinguishable, which means that two configurations are identical if they can be obtained from each other by permuting the boxes.

If you label the boxes by the number of balls they contain, then it is clear that any partition of 7, see here for partitions:

http://en.wikipedia.org/wiki/Partition_(number_theory)

defines a valid configuration and vice versa (because of the permutation symmetry). So, the answer is the number of partitions of 7, which is 15.